atmospheric moisture vapor pressure (e, pa) the partial pressure exerted by the molecules of vapor...

Atmospheric Moisture

• Vapor pressure (e, Pa) The partial pressure exerted by the

molecules of vapor in the air.

• Saturation vapor pressure (es, Pa)

The vapor pressure when in equilibrium with a plane surface of pure water. Only function of T.

L=2.5E6 J/kg, latent heat of vaporization at 0oC.

Rv= 461 JK-1kg-1, gas constant of water vapor

• Mixing ratio ( kg/kg or g/kg)

air dry of massvapor water of mass

mm

qd

v

T.RL

Pae

lnv

s 1

15273

1

611


> Saturation mixing ratio (skew-T log-P diagram)

)(622.0=

s

ss e-p

eq

)ep(e

.

R

R

)ep(e

T/R/)ep(T/R/e

T/R/pT/R/e

ρρ

V/mV/m

mm

q

v

d

d

v

dd

v

d

v

d

v

d

v

6220

,TRρe vv ,TRρp ddd

1-1-1-

-1-1

d

*

d kg K J kmol kg

kmol K J MR

R 287≈29

3.8314==

1-1-1-

-1-1

v

*

v kg K J kmol kg

kmol K J MR

R 61.54≈18

3.8314==

function of P & T

air)? moist( R dv R>R>R


• Dew point (same unit as temperature) (Td)

The temperature at which saturation would occur if moist air was cooled isobarically (at constant pressure).

xxTd T


• Wet Bulb Temperature (Tw) The lowest temperature that can be

reached by evaporating water into the air.

TTT wd

xxTd TTw

Evaporative cooling


• Dew point (same unit as temperature) (Td) The temperature at which saturation would

occur if moist air was cooled isobarically (at constant pressure)

• Wet Bulb Temperature (Tw) The lowest temperature that can be reached

by evaporating water into the air.

• Virtual temperature (Tv) Rather than use a gas constant for moist air,

which is a function of moisture, it is more convenient to retain the gas constant for dry air and define a new temperature (called virtual temperature) in the equation of the ideal gas law.

vdTRρRTρp

TTT wd

)T>T,R>R( vd


• Virtual temperature (Tv) (continue) The virtual temperature is the

temperature that dry air must have in order to have the same density as the moist air at the same pressure.

,TRρe vv ,TRρp ddd epp d +=

v

d

d

v

d

d

ddvdd

vddvd

vd

R

R

pe

TRp

R

R

Pe

pe

TRp

R/TPRReP

TPReP

TRp

TRe

TRe

TRp

TRe

TRep

ρρρ

11

1


vd

v

dd TRρ

RR

pe

TRρp

11

))kg/kg(q.(TR

R

)ep(e

T

RR

pe

TT

v

d

v

dv

610111

11

v

d

d R

R

pe

TRp

ρ 11

)ep(e

.q:Note

6220


• If T = 300 K, p = 1000 mb, and e = 15 mb, what is the value of Tv ?

• Moist air is less dense than dry air; therefore, Tv is always greater than T.

• However, even for very warm, moist air Tv exceeds T by only a few degrees.

)q.(TTv 6101

K 74.301=

))kg/kg( 0095.0×61.0+1(×300Tv ≈

kg/kg 0095.0=mb )15-1000(

mb 51×622.0=q


• Equivalent potential temperature Convert all latent heat to sensible heat

and return to 1000 mb It is approximately conserved during the

moist process.

)( eθ

How to find from a skew-T log-P diagram? eθ

eθTd

T 303 K


• Equivalent potential temperature Convert all latent heat to sensible heat

and return to 1000 mb It is approximately conserved during the

moist process.

1-1-

p

p

ve

Kg K J 1004

pressure constant at air dry of heat specific :C

,TCqL

expθθ

≈

≈

• Relative humidity (RH)

%100×ee

≈%100×qq

=RHss

Atmospheric Pressure – Altitude calculations

• Pressure is an important thermodynamic variable in itself and gradients in pressure drive the wind!

• In most places, most of the time the vertical accelerations in atmosphere are quite small, so is vertical velocity.

• As a result, the pressure distribution in the vertical is hydrostatic, i.e., at any height,

Pressure = weight of air mass above/area

orgρ

dzdp

gdzρdp

~ 1 cm/s


• Replace density with the idea gas law for dry air

• Rearranging, approximating T to Tv, and integrating from z1 to z2

• If the column is isothermal (T =constant),

air dry for constant gas the is R

T potential virtual isT where TRρp

d

vvd

2

1

2

1

p

pdz

z pdp

Tg

Rdz

gdzTRP

gdzρdpvd

2

1

2

1

p

pdz

z pdp

g

TRdz


• If the column is isothermal (T =constant),

• z2-z1 is called the thickness of the layer between p2 and p1.

• Setting p1 equal the surface pressure ps, and solving for p2 gives,

Hzz

EXPpp s12

2

1

212

2

1

2

1

pp

lng

TRzz

pdp

g

TRdz

d

p

pdz

z

-- Hypsometric equation

=> Thickness is proportional to T


• Where H is the scale height or the height in an isothermal atmosphere where the pressure has fallen to 1/e of its surface value (e-folding).

units) SI or MKS (in T.g

TRH d 329

If is 280 K, what is H?T ~ 8-9 km

1000-500 hPa thickness


• In reality, T is not a constant with height but decreases with increasing altitude in the troposphere. Hence, we need to account for this temperature variation when integrating the hydrostatic equation.

• Assuming

• Then

rate lapse )tenvironmen(dzdT

Γ

2

1

2

1 11 Γ

z

z

z

z )zz(Tdz

)z(Tdz


• We had

• Substituting and integrating to find p2 at Z2 gives:

2

1

2

1

2

1

2

1

2

1

11 Γ

z

zd

z

zd

p

p

p

pdz

z

)zz(Tdz

Rg

)z(Tdz

Rg

pdp

pdp

Tg

Rdz

=>

Γ

2

11

Γ

121

112 Γ

d

d

Rg

Rg

TT

p

)zz(TT

pp


• With these relations, p at any altitude or the altitude of any pressure level can be calculated. Typically the T lapse rates are assumed constant through discrete layers so the integration derived above is done piecewise through layers for which is approximately constant in each layer.

Γ

Z1, T1, P1

P2?, Z2, T2

Γ

2

11

Γ

121

112 Γ

d

d

Rg

Rg

TT

p

)zz(TT

pp

Γ

2

11

Γ

121

112 Γ

d

d

Rg

Rg

TT

p

)zz(TT

pp

Step1: Calculate and then P2

Step 2: Calculate and then P3

P3?, Z3, T3


• In most meteorological work, p is used as the vertical coordinate (like, 500-mb surface) rather than geometric altitude.

• If pressures are known and z of a pressure surface needs to be computed, the corollary equation for a layer with a known lapse rate is:

pT

where ,ppT

zzg

Rd

Γ1

Γ

Γ

1

2112


Potential temperature

From an Oakland sounding


• How to calculate the lapse rate at each layer?

km/C 12.76= m/C 12760.0=m 6)-(147C12.8)-(11

-=Γ ooo

km/C 8.94=m/C 89400.0=m 791)-(914

C 6)-(4.9-=Γ oo

o


Potential temperature

From an Oakland sounding

θ decreases with height,

absolutely unstable.

How to Calculate Sea Level Pressure

• Why we need sea level pressure?

• How to calculate it?

Estimate the lapse rate (can use the first few layer data from

radiosonde)

Calculate Tslp using surface height, zs, and Ts .

Then calculate pslp using ps, Ts, and Tslp .

Γ

2

112

dRg

TT

pp

)zz(TT sslpsslp Γ

ΓdRg

slp

ssslp T

Tpp

Γ

atmospheric moisture vapor pressure (e, pa) the partial pressure exerted by the molecules of vapor...

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